[Paper Review] Building reverse plane partitions with rim-hook-shaped bricks

Robin Sulzgruber|arXiv (Cornell University)|Dec 12, 2016

Advanced Mathematical Theories and Applications3 references3 citations

TL;DR

This paper presents a novel bijective construction for reverse plane partitions using rim-hook-shaped bricks as building blocks, offering an alternative to the classical Hillman–Grassl correspondence. The method systematically inserts rim-hooks into reverse plane partitions via a layered, iterative insertion process, establishing a unique lexicographic factorization that yields a bijective proof of Stanley's generating function formula for reverse plane partitions.

ABSTRACT

The generating function of reverse plane partitions of a fixed shape factors into a product featuring the hook-lengths of this shape. This result, which was first obtained by Stanley, can be explained bijectively using the Hillman-Grassl correspondence between reverse plane partitions and tableaux weighted by hook-lengths. In this extended abstract an alternative bijection between the same families of objects is presented. This construction is best perceived as a set of rules for building reverse plane partitions, viewed as arrangements of stacks of cubes, using bricks in the shape of rim-hooks.

Motivation & Objective

• To provide a new bijective proof of Stanley's generating function for reverse plane partitions, which factors into a product over hook-lengths.
• To develop a constructive, algorithmic method for building reverse plane partitions using rim-hooks as fundamental building blocks.
• To establish a unique lexicographic factorization of reverse plane partitions into rim-hooks, ensuring bijectivity between reverse plane partitions and multisets of rim-hooks.
• To offer an alternative to the Hillman–Grassl correspondence by introducing a geometric, brick-by-brick insertion process based on rim-hook shapes.

Proposed method

• The forward map constructs reverse plane partitions by iteratively inserting rim-hooks into a zero partition, using a layered insertion rule that respects the rim-hook's shape and connectivity.
• Insertion proceeds by placing the rim-hook such that its endpoints align with the shape; if it fails due to gaps, the largest possible initial segment is inserted, and the remainder is shifted diagonally and reinserted.
• The inverse map identifies the lexicographically smallest candidate cell in the partition's boundary to extract the first rim-hook, using a North-East path defined by local descent conditions.
• A path $ Q(/pi,u) $ is constructed from a candidate cell $ u $, with movement rules based on value comparisons with north and west neighbors, ensuring the path remains within the partition and respects the reverse plane partition order.
• The rim-hook $ h(/pi,u) $ is defined by its content interval and length, matching the path $ Q(/pi,u) $, and its removal results in a valid reverse plane partition if the path is $ \pi $-compatible.
• The lexicographic factorization is achieved by recursively applying the inverse map to the reduced partition, starting from the minimal candidate under content order.

Experimental results

Research questions

• RQ1 Can reverse plane partitions be uniquely reconstructed from a multiset of rim-hooks through a constructive insertion process?
• RQ2 Does a lexicographic ordering of rim-hooks yield a unique factorization of any reverse plane partition?
• RQ3 How does the new insertion algorithm compare to the Hillman–Grassl correspondence in terms of structure and combinatorial interpretation?
• RQ4 Can the trace generating function for reverse plane partitions be derived directly from this new bijective construction?

Key findings

• Every reverse plane partition of a given shape can be uniquely built by successively inserting rim-hooks in lexicographic order, establishing a bijection with multisets of rim-hooks of that shape.
• The insertion process is well-defined: if a rim-hook cannot be inserted directly, it is split into an initial segment and a shifted remainder, which are inserted in sequence.
• The inverse map successfully recovers the lexicographic factorization by identifying the minimal candidate cell under content order, ensuring uniqueness.
• The path $ Q(/pi,u) $ constructed from a candidate cell $ u $ is $ \pi $-compatible and yields a valid rim-hook factor $ h(/pi,u) $, with $ \pi - Q(\pi,u) $ remaining a reverse plane partition.
• The trace generating function is recovered via the bijection, as each rim-hook contributes a product of $ q_k $ over its content interval, matching the hook-length formula.
• The method provides an alternative bijective proof of Stanley's generating function, $ \sum_{\pi} q^{| one|} = \prod_{u \in \lambda} \frac{1}{1 - q^{h(u)}} $, through a geometric, brick-based construction.